I have not looked at this using app designer. If you were doing it outside of app designer you would have two choices:
1) go in at the Java level to adjust the properties of the Java swing object; or
2) put the text area inside a uipanel that is inside a second uipanel, and as you add text, keep making the text area larger so that it does not create scroll bars, and set the position of the inner uipanel so that the bottom of it is visible inside the outer uipanel. The outer uipanel works like a frame allowing only a fixed size of the inner panel to be visible.